CLAUDIA CHAIO AND PIOTR MALICKI
Abstract. We study the composition of irreducible morphisms between indecom-posable modules lying in quasi-tubes of the Auslander-Reiten quivers of artin algebras in relation with the powers of the radical of their module category.
1. Introduction and the main results
Throughout this paper, by an algebra we mean an artin algebra over a fixed com-mutative artin ring R. We denote by mod A the category of finitely generated right A-modules and by ind A the full subcategory of mod A consisting of one representative of each isomorphism class of indecomposable A-modules.
We denote the radical of the module category mod A by <A. We recall that, for
X, Y ∈ ind A the ideal <A(X, Y ) is the set of all non-isomorphisms between X and
Y . Inductively, the powers of <A(X, Y ) are defined. By <∞A(X, Y ) we denote the
intersection of all powers<iA(X, Y ) of<A(X, Y ) with i ≥ 1.
There is a close relationship between irreducible morphisms and the powers of the radical of its module category. In [5], Bautista proved that a morphism f : X → Y between indecomposable modules X and Y in mod A is irreducible if and only if f ∈ <A(X, Y )\<2A(X, Y ). This was generalized by Igusa and Todorov in [20, Theorem 13.3]
where they proved that, for a sectional path X0 f1 −−−→ X1 f2 −−−→ · · ·−−−→ Xfn−1 n−1 fn −−−→ Xn
of irreducible morphisms between indecomposable A-modules we have that their com-position fn. . . f2f1 ∈ <nA(X0, Xn)\ <n+1A (X0, Xn).
We denote by ΓAthe Auslander-Reiten quiver of A, and by τAand τA−1 the
Auslander-Reiten translations DTr and TrD, respectively. Recall that ΓA is a valued translation
2010 Mathematics Subject Classification. 16G70, 16G20, 16E10.
Key words and phrases. Irreducible morphism, Radical, Quasi-tube, Auslander-Reiten quiver,
Self-injective algebra.
Preprint 2015.
quiver defined as follows: the vertices of ΓAare the isomorphism classes [X] of modules
X in ind A, we put an arrow from [X]→ [Y ] in ΓA if there is an irreducible morphism
from X to Y in mod A. The valuation (dXY, d0XY) of an arrow [X]→ [Y ] in ΓAis defined such that dXY is the multiplicity of Y in the codomain of the minimal left almost split
morphism for X and d0XY is the multiplicity of X in the domain of the minimal right almost split morphism for Y . We shall not distinguish between an indecomposable A-module and the vertex of ΓA corresponding to it. Moreover, the valuation (1, 1) of
an arrow in ΓA will be omitted and we will say that a component Γ of ΓA has trivial valuation if all arrows in Γ have valuation (1, 1).
By a component of ΓAwe mean a connected component of the quiver ΓA. In general,
the Auslander-Reiten quiver ΓA describes only the quotient category mod A/<∞A.
An important research direction towards understanding the structure of module cat-egories is the study of compositions of irreducible morphisms between indecomposable modules.
In [22], S. Liu introduced the notion of degree of an irreducible morphism of mod-ules (2.3) and using such a concept he described the shapes of the components of the Auslander-Reiten quivers of algebras of infinite representation type. Liu also, studied the composition of irreducible morphisms between indecomposable modules, general-izing Igusa and Todorov result concerning sectional paths. More precisely, Liu defined the notion of pre-sectional path (2.4) and proved that if
X0 −→ X1 −→ · · · −→ Xn−1 −→ Xn
is a pre-sectional path then there are irreducible morphisms gi : Xi−1 −→ Xi for
i = 1, . . . , n, such that their composition gn. . . g2g1 lies in<nA(X0, Xn)\<n+1A (X0, Xn).
Recently, there has been many new results related to the subject of the composition of irreducible morphisms and their relation with the power of the radical of their module category. Most of them involving the concept of degree. For instance, see [7, 8, 9, 11, 12, 13, 14, 15, 16].
In [11], the authors looked at the general situation of when the composite of two irreducible morphisms is a non-zero morphism and lies in <3
A for A an artin algebra.
In particular, by [15] we are able to determine if a finite dimensional algebra over an algebraically closed field is of finite representation type by computing the degree of a finite number of irreducible morphisms. Moreover, in [9] whenever we deal with a representation-finite algebra, the minimal lower bound m≥ 1 such that <mA vanishes,
was given. This bound was determined in terms of the right and the left degree of irreducible morphisms, not depending on the maximal length of the indecomposable modules. This result was extended in [10] where the authors found the nilpotency of the radical of a module category for any artin algebra.
In [14], the authors studied the finiteness of degrees of irreducible morphisms between indecomposable modules lying in coherent almost cyclic components of Auslander-Reiten quivers of artin algebras.
In the representation theory of selfinjective algebras a prominent role played the components called quasi-tubes, whose stable parts are stable tubes. By general theory [22], [35], an infinite component Γ of the Auslander-Reiten quiver ΓA of a selfinjective
algebra A is a tube if and only if Γ contains an oriented cycle. The quasi-tubes occur in the Auslander-Reiten quivers of many selfinjective algebras, for example, for: the infinite blocks of group algebras [18], [19], the representation-infinite tame algebras [34], the selfinjective algebras of wild canonical type [21], and the deformed preprojective algebras of generalized Dynkin type [6]. We also refer to the article [31] for the bound on the number of simple and projective modules in the quasi-tubes of the Auslander-Reiten quivers of finite dimensional selfinjective algebras over a field.
We would like to mention that the quasi-tubes occur also in the Auslander-Reiten quivers of the generalized multicoil algebras (see for instance [24, 25, 26, 27, 28, 29, 30, 32] for their structure and importance), which are obtained by sophisticated gluings of concealed canonical algebras using ten admissible algebra operations, generalizing the coil operations introduced in [2].
In this paper we are interested in the composition of irreducible morphisms between indecomposable modules lying in quasi-tubes of Auslander-Reiten quivers of artin al-gebras. In particular, we study the composition of irreducible morphisms between indecomposable modules in selfinjective algebras (where projective are also injective A-modules) and tubes in a general sense.
Let A be an artin algebra. In order to formulate one of our main results we define a special type of full translation subquiver of ΓA. A full translation subquiver of ΓAof
X !! Z !! !! == !! !! == !! == !! == !! == !! == !! == !! == == !! == == !! == == !! !! == !! == !! == !! //Y // == ==
with X, Y and Z indecomposable projective-injective A-modules is said to be a special configuration of modules.
The main results proven in this work are the following theorems.
Theorem A. Let A be a selfinjective artin algebra and Γ an infinite component of ΓA
without special configurations of modules and containing an oriented cycle. Let X1 f1 −−−→ X2 f2 −−−→ · · ·−−−→ Xfn−1 n fn −−−→ Xn+1
be a path of irreducible morphisms with Xi ∈ Γ for i = 1, . . . , n + 1. Then, fn. . . f1 ∈
<n+1
A (X1, Xn+1) if and only if fn. . . f1 ∈ <∞A(X1, Xn+1).
Theorem B. Let A be an artin algebra and Γ a tube in ΓA. Let hi : Xi→Xi+1
be n irreducible morphisms with Xi ∈ Γ for i = 1, . . . , n. Then, 0 6= hn. . . h1 ∈
<n+1A (X1, Xn+1) if and only if 06= hn. . . h1 ∈ <∞A(X1, Xn+1).
For basic background on the representation theory of algebras we refer to [1], [4] and [33].
2. Preliminaries
2.1. Let A be an algebra, X, Y be the modules in ind A, and f : X → Y be an irreducible morphism in mod A. If X is not injective, we shall denote by (X) the almost split sequence starting at X and by α(X) the number of indecomposable direct summands of the middle term of (X).
2.2. Let A be an algebra. Given X, Y ∈ mod A, the ideal <A(X, Y ) is the set of
all the morphisms f : X → Y such that, for each M ∈ ind A, each h : M → X and each h0 : Y → M the composition h0f h is not an isomorphism. In particular, if X, Y ∈ ind A then <A(X, Y ) is the set of all the morphisms f : X → Y which are
not isomorphisms. Inductively, the powers of<A(X, Y ) are defined. By <∞A(X, Y ) we
denote the intersection of all powers<iA(X, Y ) of <A(X, Y ), with i≥ 1.
Next, we state the definition of degree of an irreducible morphism given by S. Liu in [22].
2.3. Let A be an algebra and let f : X → Y be an irreducible morphism in mod A, with X or Y indecomposable. Following [22] the left degree dl(f ) of f is infinite, if for each
integer n≥ 1, each module Z ∈ mod A and each morphism g ∈ <nA(Z, X)\ <n+1A (Z, X) we have that f g /∈ <n+2A (Z, Y ). Otherwise, the left degree of f is the smallest positive integer m such that there is an A-module Z and a morphism g∈ <mA(Z, X)\<m+1A (Z, X) such that f g∈ <m+2A (Z, Y ).
The right degree dr(f ) of an irreducible morphism f is dually defined.
2.4. Let A be an algebra. By a path in ΓAwe mean a sequence of irreducible morphisms between indecomposable modules Y1 → Y2 → · · · → Yn−1 → Yn, and by a non-zero
path (zero-path) we mean that the composition of the irreducible morphisms of the path does not vanish (vanishes).
In [5], Bautista defined the notion of sectional paths. A path Y1 → Y2 → · · · →
Yn−1 → Yn in ΓA is said to be sectional if for each i = 2, . . . , n− 1 we have that
Yi+16' τA−1Yi−1.
In [22], Liu generalized such a concept defining what he called a pre-sectional path. A path Y1 → Y2 → · · · → Yn−1 → Yn in ΓA is said to be pre-sectional if, whenever
Yi−1 = τAYi+1 for i = 2, . . . , n− 1 then Yi−1⊕τAYi+1 is a summand of the domain of
a right almost split morphism for Yi, or equivalently, whenever τA−1Yi−1= Yi+1 implies
that τA−1Yi−1⊕Yi+1 is a summand of the codomain of a left almost split morphism for
Yi. Observe that any sectional path is a pre-sectional path. Furthermore, in [20] Igusa and Todorov proved that if
X0 f1 −−−→ X1 f2 −−−→ · · ·−−−→ Xfn−1 n−1 fn −−−→ Xn
is a sectional path then the composition fn. . . f1 : X0 → Xn is such that fn. . . f1 ∈
<n(X
0, Xn)\<n+1(X0, Xn). In [22, Lemma 1.15], Liu extended the above result to
pre-sectional paths and proved that if X0 → X1 → · · · → Xn−1 → Xn is a pre-sectional
path then there are irreducible morphisms fi : Xi → Xi+1 for i = 0, . . . , n− 1 such
that fn−1. . . f0 ∈ <n(X0, Xn)\<n+1(X0, Xn).
By a cycle in ΓA we mean a sequence of irreducible morphisms between
indecom-posable modules of the form Y1 → Y2 → · · · → Yn−1→ Yn → Y1.
2.5. Recall that if A∞ is the quiver 0 → 1 → 2 → · · · (with trivial valuations (1,1)),
then ZA∞ is the translation quiver of the form:
(i − 1, 0) (i, 0) (i + 1, 0) (i + 2, 0) (i − 1, 1) (i, 1) (i + 1, 1) (i − 1, 2) (i, 2) @ @ R @ @ R @ @ R @ @ R @ @ R @ @ R @ @ R @ @ R @ @ R · · · · · · · · · ··· ··· ··· ···
with τ (i, j) = (i−1, j) for i ∈ Z, j ∈ N. For r ≥ 1, denote by ZA∞/(τr) the translation
quiver Γ obtained from ZA∞ by identifying each vertex (i, j) of ZA∞ with the vertex
τr(i, j) and each arrow x → y in ZA∞ with the arrow τrx → τry. The translation
quiver of the form ZA∞/(τr) is called stable tube of rank r. The rank of a stable tube
Γ is the least positive integer r such that τrx = x for all x in Γ. The τ -orbit of a stable tube Γ formed by all vertices having exactly one direct predecessor is said to be the mouth of Γ.
Let (Γ, τ ) be a translation quiver with trivial valuations. For a vertex X in Γ, called the pivot, we shall define two admissible operations [3] modifying (Γ, τ ) to a new trans-lation quiver (Γ0, τ0) depending on the shape of paths in Γ starting from X.
(ad 1) Suppose that Γ admits an infinite sectional path X = X0 → X1 → X2 → · · ·
starting at X, and assume that every sectional path in Γ starting at X is a subpath of the above path. For t≥ 1, let Γt be the following translation quiver, isomorphic to
the Auslander-Reiten quiver of the full t× t lower triangular matrix algebra, ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ❅❅❘ ❅❅❘ ❅❅❘ ❅❅❘ ❅❅❘ ❅❅❘ ❅❅❘ ❅❅❘ ❅❅❘ ❅❅❘ ❅❅❘ ✒ ✒ ✒ ✒ ✒ ✒ ✒ ✒ ✒ ✒ . . . . . . . . . .. . . . . Y1 Y2 Yt−1 Yt
We then let Γ0 be the translation quiver having as vertices those of Γ, those of Γt,
additional vertices Zij and Xi0 (where i ≥ 0, 1 ≤ j ≤ t) and having arrows as in the
figure below
The translation τ0 of Γ0 is defined as follows: τ0Zij = Zi−1,j−1 if i≥ 1, j ≥ 2, τ0Zi1 =
Xi−1 if i ≥ 1, τ0Z0j = Yj−1 if j ≥ 2, Z01 is projective, τ0X00 = Yt, τ0Xi0 = Zi−1,t if
i≥ 1, τ0(τ−1Xi) = Xi0 provided Xi is not injective in Γ, otherwise Xi0 is injective in Γ0.
For the remaining vertices of Γ0, τ0 coincides with the translation of Γ, or Γt,
sectional path consisting of the vertices Xi0, i≥ 0.
(ad 2) Suppose that Γ admits two sectional paths starting at X, one infinite and the other finite with at least one arrow
Yt← · · · ← Y2 ← Y1 ← X = X0 → X1 → X2 → · · ·
such that any sectional path starting at X is a subpath of one of these paths and X0
is injective. Then Γ0 is the translation quiver having as vertices those of Γ, additional vertices denoted by X00, Zij, Xi0 (where i≥ 1, 1 ≤ j ≤ t), and having arrows as in the
figure below
The translation τ0 of Γ0 is defined as follows: X00 is projective-injective, τ0Zij = Zi−1,j−1
if i ≥ 2, j ≥ 2, τ0Zi1 = Xi−1 if i ≥ 1, τ0Z1j = Yj−1 if j ≥ 2, τ0Xi0 = Zi−1,t if i ≥ 2, τ0X10 = Yt, τ0(τ−1Xi) = Xi0 provided Xi is not injective in Γ, otherwise Xi0 is injective
in Γ0. For the remaining vertices of Γ0, τ0 coincides with the translation τ of Γ.
We denote by (ad 1∗) and (ad 2∗) the admissible operations dual to the admissible operations (ad 1) and (ad 2), respectively.
A connected translation quiver Γ is said to be a quasi-tube if Γ can be obtained from a stable tubeT = ZA∞/(τr) by an iterated application of admissible operations
(ad 1), (ad 2), (ad 1∗) or (ad 2∗). A tube (in the sense of [17]) is a quasi-tube having the property that each admissible operation in the sequence defining it is of the form
(ad 1) or (ad 1∗), that is, it contains a cyclical path and its underlying topological space is homeomorphic to S1 × R+, where S1 is the unit circle and R+ is the nonnegative
real line. Finally, if we apply only operations of type (ad 1) (respectively, of type (ad 1∗)), then such a quasi-tube Γ is called a ray tube (respectively, a coray tube). Observe that a quasi-tube without injective (respectively, projective) vertices is a ray tube (respectively, a coray tube). A quasi-tube Γ whose all non-stable vertices are projective-injective is said to be smooth.
The following proposition provides a characterization of quasi-tubes in the Auslander-Reiten quivers of selfinjective artin algebras ([26, Theorem A], [22],[35]).
Proposition 2.6. Let A be a selfinjective artin algebra and Γ a component of ΓA. The
following statements are equivalent: (a) Γ is a quasi-tube.
(b) Γs is a stable tube.
(c) Γ contains an oriented cycle.
Here, Γs denotes the stable part of Γ, obtained from Γ by removing the projective-injective modules and the arrows attached to them.
Let A be an algebra, and let T be a stable tube of ΓA. Then T has two types of
arrows: arrows pointing to infinity and arrows pointing to the mouth. Hence, for any module Z lying in T , there is a unique sectional path X1 → X2 → · · · → Xt = Z in
T with X1 lying on the mouth of T (consisting of arrows pointing to infinity) and
there is a unique sectional path Z = Y1 → Y2 → · · · → Yt with Yt lying on the mouth
of T (consisting of arrows pointing to the mouth), and t is called the quasi-length of Z inT , denoted by ql(Z). Now, let C be a smooth quasi-tube in ΓA. Then the stable partCs ofC is a stable tube, and we may define the smooth quasi-length sql(X) of X fromC as follows:
sql(X) = ql(X) if X ∈ C
s,
ql(X+) otherwise,
where for X ∈ C \ Cs, X+ (respectively, X−) denotes the immediate successor (respectively, immediate predecessor) of X in C . Note that, if X ∈ C \ Cs then sql(X) = ql(X+) = ql(X−).
2.7. We recall that a component Γ of ΓAis called almost cyclic if all but finitely many
modules of Γ lie on oriented cycles. Further, a component Γ of ΓA is called coherent if
(C1) For each projective module P in Γ there is an infinite sectional path P = X1 → X2 → · · · → Xi → Xi+1 → Xi+2→ · · ·
(C2) For each injective module I in Γ there is an infinite sectional path · · · → Yj+2 → Yj+1 → Yj → · · · → Y2 → Y1 = I.
3. The results
We start this section recalling the definition of depth of a morphism given in [10]. Definition 3.1. Let A be an artin algebra and f : M → N be a morphism in mod A. We say that the depth of f , denoted by dp(f ), is infinite in case f ∈ <∞A(M, N ); otherwise, is the integer n≥ 0 for which f ∈ <nA(M, N ) but f /∈ <n+1A (M, N ).
For the convenience of the reader we state [8, Lemma 2.1] and [8, Proposition 2.2] which we will useful all through this paper. In fact, taking into account these results it is not hard to see that it is enough to study the irreducible morphisms satisfying the mesh relations of the components in consideration in order to have information on the irreducible morphisms of mod A.
Lemma 3.2. ([8, Lemma 2.1]) Let A be an artin algebra and Γ be a component of ΓA with trivial valuation. Let hi : Xi → Xi+1 be an irreducible morphism with Xi ∈ Γ, for
i = 1, . . . , n. Then, for any choice of irreducible morphisms fi : Xi → Xi+1 we have
that hn...h1 = δfn. . . f1+ µ with δ ∈ Aut(Xn+1) and µ∈ <n+1A (X1, Xn+1).
Let f : X → Y be an irreducible morphism between indecomposable modules in mod A. We set
Irr(X, Y ) =<A(X, Y )/<2A(X, Y ).
We recall that Irr(X, Y ) is a kX − kY−bimodule where kX = End(X)/<A(X, X) and
kY = End(Y )/<A(Y, Y ). Moreover, kZ is a division ring whenever Z is an indecom-posable A-module.
Proposition 3.3. [8, Proposition 2.2] Let A be an artin algebra and Xi ∈ ind A for
1 ≤ i ≤ n + 1. Assume that dim
kXiHomA(Xi, Xi+1) = dimkXi+1HomA(Xi, Xi+1) = 1,
for i = 1, . . . , n. Then, the following conditions are equivalent:
(a) There are irreducible morphisms fi : Xi → Xi+1 in mod A, for i = 1,· · · , n
(b) Given any irreducible morphisms hi : Xi → Xi+1 in mod A, for i = 1,· · · , n,
then hn...h1 ∈ </ n+1A (X1, Xn+1).
We shall dedicate the first part of this paper to study the composition of irreducible morphisms lying in an exceptional wing. We observe that these mentioned wings appear in coherent almost cyclic Auslander-Reiten components (see [26]). We start given the definition of exceptional wings.
Definition 3.4. A full translation subquiver of ΓA of one of the forms
## ## ## ## ## ## ;; ## ;; ## ;; ## //X1 // ;; ;; ## ;; ;; ## ... ;; ;; ;; ## ;; ## // Xn−1 // ## ;; ## ;; ## ;; ## // Xn // ;; ;; ## ## ## ## ## ## ## ## ;; ## ;; ## ;; ## ;; ## ;; ## ;; ;; ;; ## ;; ## ;; ## // X1 // ;; ;; ## ;; ;; ## ... ;; ;; ;; ## ;; ## //Xn−1 // ## ;; ## ;; ## ;; ## //Xn // ;; ;;
where n ≥ 1, Xi, 1 ≤ i ≤ n are indecomposable projective-injective modules, and
Z1 ## Z2 ## Z3 ## ## Zr−1 ## Zr ## Zr+1 ;; ## ;; ## ;; ## ;; ;; ## ;; ;; ## ... ;; ;; ;; ## ;; ## ## ;; ## ;; ## ;; ## ;; ;;
is said to be an exceptional wing. We denote it byW. The two maximal sectional paths of W are called the borders of W.
Definition 3.5. We say that a composition ϕnm. . . ϕn1 of morphisms (irreducible mor-phisms, resp.) ϕnj, for j = 1, . . . , m, in mod A (in a component Γ, resp.) behaves well whenever dp(ϕnj) = rj with rj ≥ 0 then we have that dp(ϕnm. . . ϕn1) = rm+· · · + r1.
We observe that for the proof of the converse of [7, Proposition 6.1] we do not need the hypothesis of Γ been a component of ΓA satisfying α(Γ) ≤ 2 (the number of indecomposable direct summands of the middle term of all almost split sequence is less than or equal to 2). Such a hypothesis was only necessary for the other implication. In order to make this comment clear we shall include a proof of this fact in Lemma 3.6, Statement (a).
Next, we prove three technical lemmas which will allow us to study the composition of irreducible morphisms lying in an exceptional wing W. More precisely, we shall prove that the composition of the irreducible morphisms in the borders of W behaves well.
Lemma 3.6. Let A be an artin algebra, Γ a component of ΓA, Xi ∈ Γ for i = 0, . . . , n
and n ≥ 1. Let f : Xn → Xn+1 be an irreducible morphism and assume that there is
X0 f1 τA−1X0 X1 f2 g1 >> τA−1X1 X2 g2 >> τA−1Xn−2 Xn−1 fn gn−1 >> Xn+1 Xn gn=f >>
with α(Xi) = 2 for i = 1, . . . , n−1, α(X0) = 1 and fn. . . f1 a sectional path. Moreover,
assume there is a morphism µ : X → Xn with X ∈ Γ such that dp(µ) = m for some
positive integer m and f µ∈ <m+2A (X, Xn+1). Then,
(a) The left degree of f is n and m≥ n.
(b) There exists a morphisms ϕ0 : X → X0 such that dp(ϕ0) = t, for some 0≤ t ≤
m− n, and fn. . . f1ϕ0 + µ∈ < m+1
A (X, X0).
(c) If ϕ0 is not an isomorphism then there exists a non-zero path of irreducible morphisms from X to X0 in mod A of length at most m− n.
Proof. (a) By hypothesis there exists a sectional path δ : X0 f1 −→ X1 f2 −→ · · · −→ Xn−1 fn −→ Xn
with δ = fn. . . f1. By [20] we know that δ ∈ <nA(X0, Xn)\<n+1A (X0, Xn). We also have
that f δ = 0 then we get that dl(f )≤ n.
On the other hand, since τA−1Xn−2 ⊕ Xn is the middle term of (Xn+1) by [22,
Proposition 1.6] we get that dl(f )≥ n. Hence, dl(f ) = n.
Now, since there is a morphism µ : X → Xn with X ∈ Γ such that dp(µ) = m for some positive integer m and f µ∈ <m+2A (X, Xn+1), then, dl(f ) ≤ m, that is, n ≤ m.
(b) Since dp(µ) = m and f µ∈ <m+2A (X, Xn+1) by [22, Lemma 1.2] there is a
mor-phism ϕn−1 : X → Xn−1such that ϕn−1 ∈ </ mA(X, Xn−1), gn−1ϕn−1 ∈ < m+1 A (X, τ −1 A Xn−2) and fnϕn−1+µ∈ < m+1 A (X, Xn+1). Then, fnϕn−1 =−µ+µm+1with µm+1 ∈ < m+1 A (X, Xn).
Therefore, dp(fnϕn−1) = m. Then, we infer that dp(ϕn−1) = r for some n− 1 ≤ r < m. In fact, assume that r < n− 1. Note that in such a case n > 1. If ϕn−1 is an isomor-phism and since dp(fnϕn−1) = m then dp(fn) = m but m > 1, a contradiction to the
With a similar argument as in the proof of Statement (a) we have that dl(gn−1) =
n− 1, getting a contradiction to the fact that since dp(ϕn−1) = r with r < n− 1 and gn−1ϕn−1 ∈ <m+1A (X, τA−1Xn−2) then dl(gn−1) < r < n− 1. Therefore, we prove that dp(ϕn−1) = r for some n− 1 ≤ r ≤ m − 1.
Now, since there is a morphism ϕn−1 : X → Xn−1 such that dp(ϕn−1) = r for some
n− 1 ≤ r ≤ m − 1 and gn−1ϕn−1 ∈ < m+1 A (X, τ
−1
A Xn−2) then by [22, Lemma 1.2] we
have that there is a morphism ϕn−2 : X → Xn−2 such that ϕn−2 ∈ </ m−1
A (X, Xn−2),
gn−2ϕn−2 ∈ <mA(X, τA−1Xn−3) and fn−1ϕn−2 + ϕn−1 ∈ <mA(X, Xn−1). With the same arguments as above we can show that dp(ϕn−2) = t for some n − 2 ≤ t ≤ m − 2. Moreover, fnfn−1ϕn−2 + µ ∈ < m+1 A (X, Xn+1). In fact, since fn−1ϕn−2 + ϕn−1 ∈ <m A(X, Xn) then fnfn−1ϕn−2 + fnϕn−1 ∈ < m+1 A (X, Xn+1) where fnϕn−1 = −µ + µm+1
with µm+1 ∈ <m+1A (X, Xn+1) getting that fnfn−1ϕn−2+ µ∈ < m+1
A (X, Xn+1).
Iterating the same argument and applying successively [22, Lemma 1.2] to each possible morphism ϕi : X → Xi for i = n− 3, . . . , 0 we get that there is a morphism
ϕ0 : X → X0 such that dp(ϕ0) = t, for some 0≤ t ≤ m − n, and that fn. . . f1ϕ0+ µ∈
<m+1
A (X, Xn+1). Observe that ϕ0 can be an isomorphism.
(c) Since ϕ0 is not an isomorphism then m > n and therefore 0 < t≤ m − n. By [4, VI, Proposition 7.5] there exists a non-zero path of irreducible morphisms in mod A of length at most t. Hence, we infer that there is a path of irreducible morphisms of
length at most m− n, getting the result.
Lemma 3.7. Let A be an artin algebra. Assume that there is a configuration of almost split sequences in mod A as follows
Y1 Y2 && X1 tr && t1 @@ t2 88 .. . τA−1X1 Yr 88 X2 g2 @@ τA−1X2 X3 g3 @@ τA−1Xn−2 Xn−1 gn−1 @@ τA−1Xn−1 Xn gn @@
where α(Xi) = 2 for 2 ≤ i ≤ n − 1. Suppose there exists 1 ≤ j ≤ r such that Yj
is projective and that the path δ : X1 → · · · → Xn−1 → Xn is sectional. Consider
δi : X1 → · · · → Xi a subpath of δ and gi : Xi → τA−1Xi−1 irreducible morphisms.
Then, dl(gi) =∞ for i ≥ 2. Moreover, the composition giδi behaves well for all i≥ 2.
Proof. Assume that dl(gk) < ∞, for some 2 ≤ k ≤ n − 1. By [22, Corollary 1.2] we
know that dl(g2) < · · · < dl(gn−1) < dl(gn). Hence dl(g2) < ∞. Moreover, again by
[22, Corollary 1.2] we get that dl((t1, . . . , tr)t) < ∞, but by our assumption there is an
integer 1≤ j ≤ r such that Yj is projective getting a contradiction to [22, Lemma 1.2].
Finally, note that if i≥ 2 then the composition giδibehaves well, since δiis a sectional
path and dl(gi) =∞.
Lemma 3.8. Let A be an artin algebra and Γ be a component of ΓA with trivial
valuation. Assume we have an exceptional wing W in Γ as follows
Z1 ## Z2 ## Z3 ## ## Zr−1 ## Zr ## Zr+1 ;; ## ;; ## ;; ## ;; ;; ## fr ;; ;; ## ... ;; fr−1 ;; ;; ## ;; ## ## ;; ## ;; ## ;; ## f2 ;; Y f1 ;;
and that there exists X ∈ Γ and a morphism µ : X Y such that dp(µ) = m with m > r. Moreover, assume that any path of irreducible morphisms from X Zi in Γ
of length at most m− r + 2(i − 1) is zero. Then, dp(fr. . . f1µ) = m + r.
Proof. Assume that f1µ∈ <m+2A . Since dl(f1) = r then by Lemma 3.6 (b) there exists
a morphism ϕ0 : X Z1 such that dp(ϕ0) = t for some 0 < t≤ m − r.
Observe that since m > r then m− r > 0 and X 6' Z1. Hence ϕ0 is not an
isomor-phism. By Lemma 3.6 (c), we know that there exists a non-zero path of irreducible morphisms ϕ0 : X Z1 in mod A of length at most m− r with ϕ0 ∈ </ m−r+1A (X, Z1).
We write the path ϕ0 as follows
ϕ0 : X → Y1 → Y2 → · · · → Yj → Z1.
On the other hand, if we consider i = 1 then by hypothesis any path of irreducible morphisms in Γ from X Z1 of length at most m− r is zero. Therefore, any path γ
in Γ going through the A-modules
γ : X → Y1 → Y2 → · · · → Yj → Z1
vanishes. Since we are considering Γ a component of ΓA with trivial valuation then,
by Lemma 3.2 we have that ϕ0 = δγ + µ with δ ∈ Aut(Z1) and µ ∈ <m−r+1A (X, Z1).
Hence, ϕ0 = µ with µ ∈ <m−r+1A (X, Z1) getting a contradiction to the fact that ϕ0 ∈/
<m−r+1
A (X, Z1). Therefore, dp(f1µ) = m + 1.
Iterating this procedure over all the modules Zi, for 2 ≤ i ≤ r, we get that
dp(fr. . . f1µ) = m + r.
Now, applying Proposition 3.3 and the above lemmas we get the announced result concerning compositions of irreducible morphisms of the borders of an exceptional wing.
Proposition 3.9. Let A be an artin algebra and Γ a component of ΓA with trivial
valuation. Assume we have an exceptional wing W in Γ containing a configuration of n almost split sequences with exactly three indecomposable middle terms as follows:
Y1=W1 f1 ## W2 ## ## ## ## Wr ## Y2m+1 Y2 t ;; f2 ## ;; ## ;; ## //X1 // ;; ;; ## Y2m gm ;; Y3 ;; ## ... ;; Y2m−1 gm−1 ;; ;; ## ;; ## //Xn−1 // ## ;; fm−1 ## ;; ## ;; fm ## //Xn // g2 ;; Ym+1 g1 ;; Then,
(a) The composition of irreducible morphisms in mod A between the indecomposable A-modules of the borders of W behaves well.
(b) Any composition of irreducible morphisms in W from W1 to Wj, 2 ≤ j ≤ r is
zero.
Proof. Let W be an exceptional wing in Γ. Without loss of generality, it is enough to consider an exceptional wing as in the statement with Xi 6= 0 for i = n.
(a) First, if we consider a path involving the modules Yi for i = 1, . . . , m + 1 since any such a path is sectional we get the result by [20].
Now, by Lemma 3.7 we know that dp(g1fm. . . f1) = m+1 since fm. . . f1is a sectional
path and dl(g1) =∞.
Next, we proceed as follows. If Xn−1 is projective module then dl(g2) = ∞. Hence
dp(g2g1fm. . . f1) = m + 2 since dp(g1fm. . . f1) = m + 1. Otherwise, Xn−1 = 0 and by
Lemma 3.6 (a) we have that dl(g2) = m− 1.
Assume that g2g1fm. . . f1 ∈ <m+3A (W1, Ym+2). Then, by Lemma 3.6 (c) there
ex-ists a non-zero path in mod A of length at most 2 from W1 W2. Note that any
path in Γ from W1 to W2 of length 2 is zero. In fact, observe that the only path
of length two in Γ from W1 to W2 is the path W1 = Y1 f1
−−→ Y2 t
−−→ W2 whose
ir-reducible morphisms belong to an almost split sequence with indecomposable middle term. Hence, tf1 = 0. Since the arrows of Γ have trivial valuation then by Lemma 3.2
any other path of irreducible morphisms of length two between the same modules, let say, W1 = Y1
h1
−−→ Y2 h2
−−→ W2, is such that h2h1 = δtf1 + µ with δ ∈ Aut(W2) and
µ∈ <3
A(W1, W2). Then h2h1 ∈ <3A(W1, W2). If h2h1 6= 0 then we get a contradiction to
Lemma 3.6 (c). Therefore, we prove that we can not have a non-zero path of irreducible morphisms between W1 and W2 of length at most two. Then, by Lemma 3.8 we get
that dp(g2g1fm. . . f1) = m + 2.
Analyzing the composition of each irreducible morphism gi with 3≤ i ≤ n as above
we get that dp(gn. . . g1fm. . . f1) = m + n. Finally, applying Lemma 3.8, we get the
result.
Furthermore, any composition of the form gs. . . g1fm. . . fr with 1 ≤ s ≤ m and
1≤ r ≤ m also behaves well.
(b) It is an immediate consequence of the fact that all such paths may go through the almost split sequence starting at W1 which has exactly one indecomposable middle
Proposition 3.10. Let A be an artin algebra and Γ a component of ΓA with trivial
valuation. Assume we have an exceptional wing W in Γ containing a configuration of n almost split sequences with exactly three indecomposable middle terms as follows:
W1 ## W2 ## ## ## ## Wr−1 ## Wr ## ## ;; ## ;; ## ;; ## ;; ## ;; ## ;; ;; ;; ## ;; ## ;; ## //X1 // ;; ;; ## ;; ;; ## ... ;; ;; ;; ## ;; ## //Xn−1 // ## ;; ## ;; ## ;; ## //Xn // ;; ;; Then,
(a) The composition of irreducible morphisms in mod A between the indecomposable A-modules of the borders of W behaves well.
(b) Any composition of irreducible morphisms in W from W1 to Wj, 2 ≤ j ≤ r is
zero.
Proof. Similar to the proof of Proposition 3.9.
Proposition 3.11. Let A be an artin algebra and Γ⊂ ΓAa smooth quasi-tube with only
one almost split sequence with three indecomposable middle terms. Then the following conditions hold.
(a) The composition of r ≥ 1 cycles in Γ (mod A) from a projective-injective in-decomposable module P with sql(P ) = rank (Γs) (respectively, immediate prede-cessor or sucprede-cessor of P ) behaves well.
(b) If sql(P ) < rank (Γs) then the composition of r ≥ 1 cycles in Γ (mod A) from a projective-injective indecomposable module P (respectively, immediate prede-cessor or sucprede-cessor of P ) is zero.
Proof. Let A be an artin algebra, Γ be a smooth quasi-tube in ΓA, and P be a
first that, it follows from [26, Lemma 4.9] (see also [23, Lemmas 2.5-2.8 and their duals]) that sql(P )≤ rank (Γs).
(a) We illustrate the situation of this statement with the following diagram
X X X .. . AA AA AA AA AA AA .. . AA AA AA AA AA fm AA .. . .. . AA AA ... AA AA ... AA AA ... AA AA ... .. . AA AA AA ... AA AA AA ... .. . AA AA AA ... AA AA AA ... .. . AA AA AA ... AA AA AA ... .. . AA AA AA //P // AA ... AA AA //P // f2 AA ... .. . AA AA Y AA AA AA AA Y f1 AA AA .. . ... AA .. . AA .. . AA .. . AA .. . ... ... ... AA .. . AA .. . AA .. . AA .. . ... By Proposition 3.9 we know that the composition of the morphisms in the borders δ1
starting at X and ending at Y and δ2 starting at Y and ending at X of the exceptional
wing, behaves well. Moreover, f1δ1δ2δ1 also behaves well since dl(f1) = ∞ and the
irreducible morphisms of the border δ1 have infinite left degree. By Lemma 3.8, we
know that fm. . . f1δ1δ2δ1 = δ2δ1δ2δ1 behaves well.
Repeating this argument we get the result for Γ. Moreover, by Lemma 3.2 we get the result for mod A.
(b) Let P be a projective-injective module, sql(P ) < rank (Γs) and Xm → Xm−1 → · · · → X1 → X0 = P
be the unique sectional path in Γ with Xm lying on the mouth of Γs. Then m = sql(P ).
Observe that the smooth quasi-tube Γ is a coherent component of ΓA (2.7). Since Γ is
also a cyclic component of ΓA, applying [26, Theorem A] (see also [23, Theorem 2.3]),
we infer that Γ, considered as a translation quiver, can be obtained from a stable tube by an iterated application of admissible operations of type (ad 1), (ad 2), (ad 1∗) and (ad 2∗), described in Section 2. Moreover, by our assumption on the number of almost split sequences with three middle terms, we can apply only one admissible operation of type (ad 2) or (ad 2∗). Then it follows that HomA(P, Xi) = 0 and HomA(P, Yj) = 0,
where
is the unique infinite sectional path in Γ consisting of arrows pointing to infinity and 1≤ i≤ m, j ≥ 1. Therefore, HomA(P, X0) = HomA(P, P ) = 0 and then the composition
of r ≥ 1 cycles in Γ (mod A) from P (respectively, from immediate predecessor or
successor of P ) is zero.
Proposition 3.12. Let A be an artin algebra and Γ a smooth quasi-tube in ΓA.
As-sume we have an exceptional wing W in Γ containing a configuration of n almost split sequences with exactly three indecomposable middle terms as on the figures in Definition 3.4. Then the following conditions hold.
(a) The composition of r ≥ 1 cycles in Γ (mod A) from a projective-injective inde-composable module Xn with sql(Xn) = rank (Γs) (respectively, immediate
pre-decessor or successor of Xn) behaves well.
(b) If sql(Xn) < rank (Γs) then the composition of r ≥ 1 cycles in Γ (mod A)
from a projective-injective indecomposable module Xn (respectively, immediate
predecessor or successor of Xn) is zero.
Proof. Similar to the proof of Proposition 3.11 using additionally induction on the
number of projective-injective modules.
The next result shall be useful for further purposes.
Lemma 3.13. Let A be a selfinjective artin algebra and Γ ⊂ ΓA be a quasi-tube. Assume we have in Γ a zero path of irreducible morphisms X1 → X2 → · · · → Xn →
Xn+1. Then, any longest path in Γ from X1 Xn+1 vanishes.
Proof. Let A be a selfinjective artin algebra and Γ be a quasi-tube in ΓA. It follows from
Proposition 2.6 that the stable part Γs of Γ is a stable tube. Moreover, observe that the quasi-tube Γ is a coherent component of ΓA, that is, the following two conditions
are satisfied:
(C1) For each projective module P in Γ there is an infinite sectional path P = U1 →
U2 → U3 → · · · starting at P ;
(C2) For each injective module I in Γ there is an infinite sectional path· · · → V3 →
V2 → V1 = I ending at I.
Since Γ is also a cyclic component of ΓA, applying [26, Theorem A] (see also [23,
Theorem 2.3]), we infer that Γ, considered as a translation quiver, can be obtained from a stable tube by an iterated application of admissible operations of type (ad 1),
(ad 2), (ad 1∗) and (ad 2∗). Since the projectives and injectives vertices in Γ coincide, the projective-injective vertices in Γ are created as follows:
• for each operation (ad 1) with pivot X0 and t = 0, the operation (ad 1∗) with
pivot at X00 and t = 0 is applied;
• for each operation (ad 1∗) with pivot X
0 and t = 0, the operation (ad 1) with
pivot at X00 and t = 0 is applied;
• for each operation (ad 1) with pivot X0 and t≥ 1, the operation (ad 2∗) with
pivot at Z01 is applied;
• for each operation (ad 1∗) with pivot X
0 and t≥ 1, the operation (ad 2) with
pivot at Z01 is applied.
Now, let α : X1 → X2 → · · · → Xn → Xn+1 be a zero path of irreducible morphisms
in Γ and i be the largest index such that a subpath β : X1 → X2 → · · · → Xi of α is
non-zero. Moreover, let Zp → Zp−1→ · · · → Z1 → Xi be the unique maximal sectional
path in Γ starting at Zp and formed by arrows pointing to the infinity. Then Zp lies
on the mouth of Γs. Then, it follows from the definition of admissible operations of types (ad 1), (ad 2), (ad 1∗), (ad 2∗) that, if X1 → · · · → Y is a non-zero path of
irreducible morphisms in Γ then Y lies in the infinite rectangle S (X1, Zp) consisting
of the vertices bounded by:
• the infinite sectional path in Γ starting at X1 and formed by arrows pointing
to the infinity;
• the finite sectional path in Γ starting at X1 and formed by arrows pointing to
the mouth;
• the infinite sectional path Zp → · · · → Z1 → Xi → · · · in Γ starting at Zp and
formed by arrows pointing to the infinity.
Therefore, any longest path in Γ from X1 to Xn+1 vanishes.
Our next result shows that if A is a selfinjective artin algebra and Γ an infinite component of ΓAwithout special configurations of modules and containing an oriented
cycle then the composition of irreducible morphisms fn. . . f1 ∈ <n+1A (X1, Xn+1) if and
only if fn. . . f1 ∈ <∞A(X1, Xn+1). To achive to such a result we start proving the
following lemma.
Lemma 3.14. Let A be a selfinjective artin algebra and Γ be a quasi-tube in ΓA with
at least two projective-injective modules and such that all projective-injectives belong to exactly two exceptional wingsW and W0 in Γ. Let α : X → · · · → Y , β : Y → · · · → Z
be the borders of W and γ : U → · · · → V , δ : V → · · · → W be the borders of W0. Then the following conditions hold.
(a) If Z = U (respectively, W = X) then the composition of irreducible morphisms from X to W (respectively, from U to Z) behaves well.
(b) If Z 6= U (respectively, W 6= X) then any composition of irreducible morphisms from X to W (respectively, from U to Z) is zero.
Proof. Observe first that, it follows from [26, Lemma 4.9] (see also [23, Lemmas 2.5-2.8]) that there is in Γ the infinite rectangleS (Y, Z) consisting of the vertices bounded by:
• the infinite sectional path in Γ starting at Y and formed by arrows pointing to the infinity;
• the finite sectional path β : Y → · · · → Z in Γ;
• the infinite sectional path in Γ starting at Z and formed by arrows pointing to the infinity.
Moreover, all meshes inS (Y, Z) are with exactly two middle terms and for any T from S (Y, Z) we have HomA(X, T )6= 0.
Let Z = U . By Proposition 3.9 we know that the composition of irreducible mor-phisms of the borders of the exceptional wings behaves well. Let σ be the sectional path in Γ from infinity to V and % be the sectional path in Γ from Y to infinity. Then σ intersects % and denote by N their common module. Note that every composition of irreducible morphisms from Y to V in the rectangle S (Y, Z, V, N) is equal and non-zero. Therefore, the composition of irreducible morphisms from X to W behaves well.
If Z 6= U, then for the infinite sectional path τA−Z = M1 → M2 → · · · formed by
arrows pointing to the infinity we have HomA(X, Mi) = 0 where i ≥ 1 and Z is the
starting vertex of a mesh with exactly one middle term. Hence we get (b). Theorem 3.15. Let A be a selfinjective artin algebra and Γ a quasi-tube of ΓAwithout
special configurations of modules. Let X1 f1 −−−→ X2 f2 −−−→ · · ·−−−→ Xfn−1 n fn −−−→ Xn+1
be a path of irreducible morphisms with Xi ∈ Γ for i = 1, . . . , n + 1. Then, fn. . . f1 ∈
<n+1
Proof. We only prove that, if fn. . . f1 ∈ <n+1A (X1, Xn+1) then fn. . . f1 ∈ <∞A(X1, Xn+1)
since the other implication is clear.
To analyze the composition of irreducible morphisms in Γ we will start with the ones near the mouth of Γ. It is enough to study that all non-zero compositions behaves well. We will do induction on the number n + m with n, m≥ 1, where n is the number of exceptional wings in Γ and m is the number of projective-injective vertices from almost split sequences with exactly two middle terms in Γ. We would like to note that on the figures below we present the first exceptional wing from Definition 3.4 but for the second exceptional wing containing the meshes with exactly three middle terms, the proof is the same. Let n + m = 2, we have three cases.
(a) If n = 2 then by Lemma 3.14 we get the result.
(b) Let n = 1 and m = 1. Let P be a projective-injective module in Γ belonging to a mesh with exactly two middle terms. Consider an exceptional wing W with the borders ϕ1 starting at X and ϕ2 ending in Y . Then, the only non-zero composition of
irreducible morphisms from X to Z is gs...g1ϕ2ϕ1, where gs...g1 belong to the unique
infinite sectional path in Γ starting at Y and passing through Z (formed by arrows pointing to the infinity), and ϕ2ϕ1 is the composition of the borders of the wing W.
We illustrate the situation with the following diagram:
P .. . X Y g 1 AA ... .. . AA AA AA // // AA AA g2 AA AA AA AA .. . .. . AA AA AA AA AA AA .. . .. . AA AA AA gs ... .. . AA AA AA // // AA Z .. . .. . AA AA AA AA .. . .. . AA AA AA AA ... ... .. . AA AA AA AA AA ... .. . ... AA .. . AA .. . AA .. . AA .. . ... ... ...
In fact, by Proposition 3.9 or Proposition 3.10 the composition ϕ2ϕ1 of the borders
of the exceptional wing W behaves well. Now, since the left degree of the morphisms g1, . . . , gs are infinite then the composition gs...g1ϕ2ϕ1 behaves well. Now, consider the
situation illustrated with the following diagram:
P .. . X Y g1 AA ... .. . AA AA AA // // AA AA AA AA AA AA .. . .. . AA AA AA AA AA AA .. . .. . AA AA AA ... .. . AA AA V AA f p // q// U AA Z .. . .. . AA AA N g AA AA .. . .. . AA AA AA AA ... ... .. . AA AA AA AA AA ... .. . ... AA .. . AA .. . AA .. . AA .. . ... ... ...
Denote by ϕ the unique sectional path from X to V , by ψ the unique sectional path from U to Y , and by η the unique sectional path from P to τA−Z. Note that the compositions ηg1ψqpϕ and ηg1ψgf ϕ behaves well, since by [22] dl(g1) =∞ and by [14]
the irreducible morphisms in η have infinite left degree. Therefore, the only non-zero composition of irreducible morphisms from X to τA−Z passing through P behaves well.
(c) Let m = 2 and X, Y be the projective-injective modules in Γ. In this case, if HomA(X, Z)6= 0 (respectively, HomA(Y, Z)6= 0) then Z belongs to the unique infinite
sectional path starting at X (respectively, at Y ). Moreover, it follows by Lemma 3.16 that any path from X to Y is zero. The non-zero paths are the ones which involves almost split sequences not going through (modulo mesh) the almost split sequences with only one indecomposable middle term. In fact, this follows because one can write
such a composition as a chain of irreducible morphisms of a coray followed by a chain of irreducible morphisms in a ray. By [14] the right degree of the ones in the coray are infinite and the left degree of the ones in the ray are infinite.
Assume that for n + m− 1 the result is true. We want to prove our theorem for n + m. We have two cases:
Case 1. We fix a configuration of almost split sequences of an exceptional wing W as follows: Y1 f1 !! Z2 !! Z3 !! !! !! !! Z Y2 == f2 !! t2 == !! == !! // X1 // == == !! gs == Y3 t1 == !! ... == == X µ CC == !! == !! //X s−1 // !! == !! == !! == fs !! // Xs // g2 == Ys+1 g1 ==
Since the left degrees of f1, . . . , fs, g1 are infinite then g1fs. . . f1µ behaves well. Now
we proceed as in the proof of Proposition 3.9, that is, if Xs−1 is projective-injective then dl(g2) = ∞ and hence g2g1fs. . . f1µ behaves well. Otherwise, if Xs−1 = 0 then
by Lemma 3.8 we know that any path in Γ from Y1 to Z3, say ϕ1 : Y1 Z3, is zero.
Therefore, clearly, any longest path as t2t1f2f1µ is also zero. Iterating this procedure,
we get that the composition gs. . . g2g1fs. . . f1µ behaves well. On the other hand, if
µ : X → Yi for 2 ≤ i ≤ s then by inductive hypothesis we have that µ behaves well. Then, we have to consider the last configuration next to W. We have two cases to consider.
W Y1 Z AA AA // V1 // AA AA Y2 AA AA //X1 // AA AA AA X µ HH .. . AA Y3 AA ... AA AA AA AA ... AA AA AA //Vn // AA AA V AA AA //Xs // Ys+1 AA
Then it follows from the inductive hypothesis, Lemma 3.14 and its proof that the composition λ2λ1σ2σ1µ behaves well, where σ1 : W → · · · → V , σ2 : V → · · · → Y1,
λ1 : Y1 → · · · → Ys+1, λ2 : Ys+1 → · · · → Z are the borders of the above exceptional
wings, and S (Y1, Ys+1) is the infinite rectangle consisting of the vertices bounded by:
• the infinite sectional path in Γ ending at Y1 and formed by arrows pointing to
the mouth;
• the finite sectional path Y1 → · · · → Ys+1 in Γ;
• the infinite sectional path in Γ ending at Ys+1 and formed by arrows pointing
to the mouth.
Moreover, all meshes in S (Y1, Ys+1) are with exactly two middle terms and for any U
fromS (Y1, Ys+1) we have HomA(U, Z)6= 0.
P h ?? Y1 f1 Z ?? ?? Y2 ?? f2 ?? ?? //X1 // ?? ?? gs ?? Y3 ?? ... ?? ?? X µ 22 ?? ?? // // ?? ?? ?? fs //Xs // g2 ?? Ys+1 g1 ??
with the exceptional wing W and projective-injective module P in Γ belonging to a mesh with exactly two middle terms. By the previous considerations it is enough to consider the composition gs. . . g1fs. . . f1hµ. By the inductive hypothesis µ : X P
behaves well. Since the left degree of irreducible morphisms h, f1, . . . , fs are infinite
then the composition fs...f1hµ behaves well. Finally, also gs. . . g1fs. . . f1hµ behaves
well.
Case 2. Assume we have the following situation:
P !! !! == !! !! !! !! == !! == == !! == !! !! == == !! == == !! !! δ == == == !! V t !! == == !! == X µ 44 U == ==
where δ : V → P is a sectional path in Γ and P is a projective-injective module belonging to a mesh with exactly two middle terms. By inductive hypothesis we know that µ behaves well. The irreducible morphisms in δ have infinite left degree by [14]. Hence, δµ behaves well. Moreover, again by [14] since t is an irreducible monomorphism then dl(t) =∞ and we get the result.
Then, it is enough to prove the result for zero paths in Γ, since if we have a non-zero path X1 f1 −−−→ X2 f2 −−−→ · · · fn−1 −−−→ Xn fn −−−→ Xn+1
in Γ then, as we see above, fn. . . f1 behaves well, getting a contradiction with our
assumption. Therefore, fn. . . f1 = 0.
Now, any other composition of irreducible morphisms hi : Xi → Xi+1for i = 1, . . . , n
is such that hn. . . h1 = δfn. . . f1 + µ with µ ∈ <n+1A (X1, Xn+1) and δ ∈ Aut(Xn+1).
Hence, hn. . . h1 ∈ <n+1A (X1, Xn+1).
Assume that hn. . . h1 ∈ </ ∞A(X1, Xn+1), that is, the composition hn. . . h1 belongs to
<m
A(X1, Xn+1)\<m+1A (X1, Xn+1) with m > n. Hence there is a non-zero path from X1
Xn+1 of length longest than n, contradicting Lemma 3.13. The proof is completed.
Note that as an immediate consequence of Proposition 2.6 and Theorem 3.15 we obtain Theorem A.
Our next two results are fundamental for the study of the composition of irreducible morphisms lying in a tube.
Lemma 3.16. Let A be an artin algebra and Γ a tube in ΓA. Then
(a) If there is a zero path in Γ from X to Y then any longest path in Γ from X to Y vanishes.
(b) If there is a non-zero path γ from X to Y in Γ of length m then dp(γ) = m.
Proof. (a) Let A be an artin algebra and Γ be a tube in ΓA. From the definition of
a tube we know that Γ considered as a translation quiver can be obtained from a stable tube by an iterated application of admissible operations of type (ad 1) and (ad 1∗). Therefore, the statement follows from arguments similar to those applied in the proof of Lemma 3.13.
(b) Let γ : X = X1 → X2 → · · · → Xm → Xm+1 = Y be a non-zero path in Γ of
length m. Then we have in Γ the rectangleS (X, Zp, Y, W ) of the form
Zp "" ◦ << ## ◦ ## ;; ◦ "" ;; << ◦ >> "" ◦ X >> ◦ << "" ◦ >> Y "" << ◦ >> << ◦ >> >> "" >> ## ◦ << ## ◦ ;; ## ◦ ## ;; ;; W ;; "" <<
Observe that in this case any path in S (X, Zp, Y, W ) from X to Y is non-zero and
has length m. Let f : X → Zp be the composition of irreducible maps corresponding
to the arrows of the sectional path α : X = X1 → · · · → Zp, and let g : Zp → Y
be the composition of irreducible maps corresponding to the arrows of the sectional path β : Zp → · · · → Xm+1 = Y . Since by [22, Section 1] the arrows of the path α
(respectively, the path β) are of infinite right (respectively, left) degree, we infer that gf ∈ <mA(X, Y )\ <m+1A (X, Y ). Hence dp(γ) = m. Lemma 3.17. Let A be an artin algebra and Γ a tube in ΓA. Let hi : Xi → Xi+1 be ir-reducible morphisms with Xi ∈ Γ for i = 1, . . . , n+1. If 0 6= hn. . . h1 ∈ <n+1A (X1, Xn+1)
then there exists f1, . . . , fn such that fn. . . f1 = 0 for any choice of irreducible
mor-phisms fi : Xi → Xi+1 satisfying the mesh relations of Γ.
Proof. Consider irreducible morphisms fi : Xi → Xi+1 for i = 1, . . . , n satisfying
the mesh relations of Γ. By Lemma 3.2 we have that hn. . . h1 = δfn. . . f1+ µ with
Suppose that fn. . . f1 6= 0. Then, by Lemma 3.16 (b), we get that fn. . . f1 behaves
well getting a contradiction with the fact that fn. . . f1 ∈ <n+1A (X1, Xn+1). Hence
fn. . . f1 = 0.
Next, we prove one of our main results. We observe that the proof is similar to [8, Theorem A]. For the convenience of the reader we state it here.
Theorem 3.18. Let A be an artin algebra and Γ a tube in ΓA. Let hi : Xi→Xi+1
be n irreducible morphisms with Xi ∈ Γ for i = 1, . . . , n. Then, 0 6= hn. . . h1 ∈
<n+1
A (X1, Xn+1) if and only if 06= hn. . . h1 ∈ <∞A(X1, Xn+1).
Proof. Assume that there are n irreducible morphisms hi : Xi→Xi+1 such that 0 6=
hn. . . h1 ∈ <n+1A (X1, Xn+1). By Lemma 3.17 there are n irreducible morphisms fi :
Xi → Xi+1 in the mesh satisfying that fn...f1 = 0.
Suppose that hn. . . h1 ∈ <n+kA (X1, Xn+1)\<n+k+1A (X1, Xn+1), for some k≥ 1. By [4,
V, Proposition 7.4] there is a non-zero path γ : X1 → Xn+1 of irreducible morphisms
of length n + k, whose composition does not belong to <n+k+1A (X1, Xn+1). Then,
by Lemma 3.16 (a), we know that there is a zero path γ0 : X1 → Xn+1 of length
n + k satisfying the mesh relations in Γ. By Lemma 3.2 we can write γ0 = δγ + µ with δ ∈ Aut(Xn+1) and µ ∈ <n+k+1A (X1, Xn+1). Hence, we conclude that γ0 ∈
<n+k+1A (X1, Xn+1) a contradiction.
The converse is clear.
Acknowledgements
This research project was started when the authors visited the Mathematisches Forschungsinstitut Oberwolfach through the programme ”Research in Pairs” in 2014.
The first author acknowledge partial support from CONICET and Universidad Na-cional de Mar del Plata.
The research of the second named author has been supported by the research grant DEC-2011/02/A/ST1/00216 of the Polish National Science Center.
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Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Funes
3350, Universidad Nacional de Mar del Plata, 7600 Mar del Plata, Argentina E-mail address: claudia.chaio@gmail.com
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University,
Chopina 12/18, 87-100 Toru´n, Poland
